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19 pages, 387 KiB

Open AccessFeature PaperArticle

Logarithmic SAT Solution with Membrane Computing

by Radu Nicolescu, Michael J. Dinneen, James Cooper, Alec Henderson and Yezhou Liu

Axioms 2022, 11(2), 66; https://doi.org/10.3390/axioms11020066 - 8 Feb 2022

Cited by 1 | Viewed by 2583

Abstract

P systems have been known to provide efficient polynomial (often linear) deterministic solutions to hard problems. In particular, cP systems have been shown to provide very crisp and efficient solutions to such problems, which are typically linear with small coefficients. Building on a [...] Read more.

P systems have been known to provide efficient polynomial (often linear) deterministic solutions to hard problems. In particular, cP systems have been shown to provide very crisp and efficient solutions to such problems, which are typically linear with small coefficients. Building on a recent result by Henderson et al., which solves SAT in square-root-sublinear time, this paper proposes an orders-of-magnitude-faster solution, running in logarithmic time, and using a small fixed-sized alphabet and ruleset (25 rules). To the best of our knowledge, this is the fastest deterministic solution across all extant P system variants. Like all other cP solutions, it is a complete solution that is not a member of a uniform family (and thus does not require any preprocessing). Consequently, according to another reduction result by Henderson et al., cP systems can also solve k-colouring and several other NP-complete problems in logarithmic time. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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17 pages, 383 KiB

Open AccessArticle

A Hypergraph Model for Communication Patterns

by Gabriel Ciobanu

Axioms 2022, 11(1), 8; https://doi.org/10.3390/axioms11010008 - 23 Dec 2021

Viewed by 2414

Abstract

The article deals with interaction in concurrent systems. A calculus able to express specific communication patterns is defined, together with its abstract control structures. A hypergraph model for these structures is presented. The hypergraphs are able to properly express the communication patterns, providing [...] Read more.

The article deals with interaction in concurrent systems. A calculus able to express specific communication patterns is defined, together with its abstract control structures. A hypergraph model for these structures is presented. The hypergraphs are able to properly express the communication patterns, providing a fully abstract model for the pattern calculus. It is also proved that the hypergraph model preserves the operational reductions of processes from pattern calculus and of the actions from the control structures. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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14 pages, 330 KiB

Open AccessArticle

Two Extensions of Cover Automata

by Cezar Câmpeanu

Axioms 2021, 10(4), 338; https://doi.org/10.3390/axioms10040338 - 10 Dec 2021

Viewed by 2138

Abstract

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get [...] Read more.

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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11 pages, 273 KiB

Open AccessArticle

List Approximation for Increasing Kolmogorov Complexity

by Marius Zimand

Axioms 2021, 10(4), 334; https://doi.org/10.3390/axioms10040334 - 7 Dec 2021

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Abstract

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer [...] Read more.

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that takes as input a string x of length n and returns a list with

O (n^{2})

strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than

n - log log n - O (1)

. We also present an algorithm that obtains a list of quasi-polynomial size in which each element can be produced in polynomial time. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

12 pages, 295 KiB

Open AccessArticle

P Systems with Evolutional Communication and Division Rules

by David Orellana-Martín, Luis Valencia-Cabrera and Mario J. Pérez-Jiménez

Axioms 2021, 10(4), 327; https://doi.org/10.3390/axioms10040327 - 30 Nov 2021

Cited by 2 | Viewed by 2129

Abstract

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane [...] Read more.

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane systems can give them the ability to efficiently solve presumably intractable problems. These ingredients are called to form a frontier of efficiency, in the sense that passing from the first type of P systems to the second type leads to passing from non-efficiency to the presumed efficiency. In this work, a solution to the SAT problem, a well-known NP-complete problem, is obtained by means of a family of recognizer P systems with evolutional symport/antiport rules of length at most (2,1) and division rules where the environment plays a passive role; that is, P systems from

\hat{CDEC} (2, 1)

. This result is comparable to the one obtained in the tissue-like counterpart, and gives a glance of a parallelism and the non-evolutionary membrane systems with symport/antiport rules. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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18 pages, 383 KiB

Open AccessArticle

The Maximal Complexity of Quasiperiodic Infinite Words

by Ludwig Staiger

Axioms 2021, 10(4), 306; https://doi.org/10.3390/axioms10040306 - 17 Nov 2021

Viewed by 1372

Abstract

A quasiperiod of a finite or infinite string is a word whose occurrences cover every part of the string. An infinite string is referred to as quasiperiodic if it has a quasiperiod. We present a characterisation of the set of infinite strings having [...] Read more.

A quasiperiod of a finite or infinite string is a word whose occurrences cover every part of the string. An infinite string is referred to as quasiperiodic if it has a quasiperiod. We present a characterisation of the set of infinite strings having a certain word q as quasiperiod via a finite language

P_{q}

consisting of prefixes of the quasiperiod q. It turns out its star root

\sqrt[*]{P_{q}}

is a suffix code having a bounded delay of decipherability. This allows us to calculate the maximal subword (or factor) complexity of quasiperiodic infinite strings having quasiperiod q and further to derive that maximally complex quasiperiodic infinite strings have quasiperiods

a b a

or

a a b a a

. It is shown that, for every length

l \geq 3

, a word of the form

a^{n} b a^{n}

(or

a^{n} b b a^{n}

if l is even) generates the most complex infinite string having this word as quasiperiod. We give the exact ordering of the lengths l with respect to the achievable complexity among all words of length l. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

15 pages, 281 KiB

Open AccessArticle

On Turing Machines Deciding According to the Shortest Computations

by Florin Manea

Axioms 2021, 10(4), 304; https://doi.org/10.3390/axioms10040304 - 13 Nov 2021

Viewed by 1579

Abstract

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations [...] Read more.

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations on that word is accepting; on the other hand, the machine rejects the input word when all the shortest computations performed by the machine on that word are rejecting. We are able to show that the class of languages decided in polynomial time by such machines is

P^{NP [log]}

. When we consider machines that decide a word according to the decision taken by the lexicographically first shortest computation, we obtain a new characterization of

P^{NP}

. A series of other ways of deciding a language with respect to the shortest computations of a Turing machine are also discussed. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

6 pages, 301 KiB

Open AccessArticle

Interdimensionality

by Karl Svozil

Axioms 2021, 10(4), 300; https://doi.org/10.3390/axioms10040300 - 12 Nov 2021

Viewed by 2521

Abstract

In this speculative analysis, interdimensionality is introduced as the (co)existence of universes embedded into larger ones. These interdimensional universes may be isolated or intertwined, suggesting a variety of interdimensional intrinsic phenomena that can only be understood in terms of the outer, extrinsic reality. [...] Read more.

In this speculative analysis, interdimensionality is introduced as the (co)existence of universes embedded into larger ones. These interdimensional universes may be isolated or intertwined, suggesting a variety of interdimensional intrinsic phenomena that can only be understood in terms of the outer, extrinsic reality. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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10 pages, 244 KiB

Open AccessArticle

Simulations between Network Topologies in Networks of Evolutionary Processors

by José Ángel Sánchez Martín and Victor Mitrana

Axioms 2021, 10(3), 183; https://doi.org/10.3390/axioms10030183 - 11 Aug 2021

Cited by 1 | Viewed by 1620

Abstract

In this paper, we propose direct simulations between a given network of evolutionary processors with an arbitrary topology of the underlying graph and a network of evolutionary processors with underlying graphs—that is, a complete graph, a star graph and a grid graph, respectively. [...] Read more.

In this paper, we propose direct simulations between a given network of evolutionary processors with an arbitrary topology of the underlying graph and a network of evolutionary processors with underlying graphs—that is, a complete graph, a star graph and a grid graph, respectively. All of these simulations are time complexity preserving—namely, each computational step in the given network is simulated by a constant number of computational steps in the constructed network. These results might be used to efficiently convert a solution of a problem based on networks of evolutionary processors provided that the underlying graph of the solution is not desired. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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10 pages, 236 KiB

Open AccessCommentary

Solomon Marcus Contributions to Theoretical Computer Science and Applications

by Cristian S. Calude and Gheorghe Păun

Axioms 2021, 10(2), 54; https://doi.org/10.3390/axioms10020054 - 5 Apr 2021

Cited by 2 | Viewed by 2509

Abstract

Solomon Marcus (1925–2016) was one of the founders of the Romanian theoretical computer science. His pioneering contributions to automata and formal language theories, mathematical linguistics and natural computing have been widely recognised internationally. In this paper we briefly present his publications in theoretical [...] Read more.

Solomon Marcus (1925–2016) was one of the founders of the Romanian theoretical computer science. His pioneering contributions to automata and formal language theories, mathematical linguistics and natural computing have been widely recognised internationally. In this paper we briefly present his publications in theoretical computer science and related areas, which consist in almost ninety papers. Finally we present a selection of ten Marcus books in these areas. Full article

(This article belongs to the Special Issue In Memoriam, Solomon Marcus)

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